Symbol in context

$r$

This entry links every published equation using this exact notation. Its meaning may change between equations.

Meanings in context

Editorial equation guide

Real-valued model value

One numerical weight or activation as the model uses it. “Real-valued” means it belongs on the continuous number line: it can be negative, zero, fractional, or positive. In a computer it is stored with finite precision, so the stored float is only an approximation to a mathematical real number.

Read equation 1 in Edge AI Electronics and Sensor Systems in Practice: Quantization, Power Budgets, and Sensor Front Ends →

Article-grounded definition

Symbol r

the real-valued number.

Read equation 2 in Edge AI Electronics and Sensor Systems in Practice: Quantization, Power Budgets, and Sensor Front Ends →

Article-grounded definition

Symbol r

the radius.

Read equation 72 in A Horizon Is a Toll Booth, Not a Loophole →

Article-grounded definition

Symbol r

each model’s cache-read rate: 1.

Read equation 2 in Astra's 63%-Cheaper Claim Has to Beat a Price Disadvantage Nobody Mentions →

Article-grounded definition

Symbol r

non-trivial), and a mantle viscosity $\eta$ low enough (i.

Read equation 7 in From Origins to Frontier: A History of Planetary Science and Exploration →

Article-grounded definition

Symbol r

the coefficient of relatedness between the actor and the recipient.

Read equation 1 in Why We Punish Strangers: The Evolution of Human Cooperation →

Article-grounded definition

Symbol r

effectively zero.

Read equation 5 in Why We Punish Strangers: The Evolution of Human Cooperation →

Article-grounded definition

Symbol r

the overall reaction rate.

Read equation 1 in Chemical Dynamics and Catalysis in Practice: An Advanced Technical Guide →

Article-grounded definition

Symbol r

the distance.

Read equation 11 in The Logistics Tail: Sustainment as the Binding Constraint on Organised Force →

Article-grounded definition

Symbol r

the five counts in that order and each p is the corresponding published rate.

Read equation 1 in The Unit Economics of a Token →

Article-grounded definition

Symbol r

set by the thermal wall, and r falls as the device heats — so a long deliberation is charged at a worsening exchange rate the longer it runs, which is exactly the behaviour the sustained-load measurements show [ 6 , 5 ].

Read equation 10 in The Deployment Envelope: Small Models Where the Power Is Not →

Used in 95 equations

r(n) = c^{\,n}.

Ten Failure Modes That Define Production AI Agent Architectures · Equation 3

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Equation guide → · This term → · Article →

r

The Real Cost of Running Claude in Production, Beyond the Per-Token Price · Equation 3

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

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C = \frac{p_{\mathrm{in}}(n + \tau) + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}}(y + z)}{a}

The Real Cost of Running Claude in Production, Beyond the Per-Token Price · Equation 12

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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J(\theta) = \mathbb{E}_{x \sim \mathcal{D},\ y \sim \pi_\theta(\cdot \mid x)}\left[r(x,y)\right] - \beta D_{\mathrm{KL}}\left(\pi_\theta(\cdot \mid x) \,\|\, \pi_{\mathrm{ref}}(\cdot \mid x)\right)

Claude, From First Principles: Training, Constitutional Methods, and What Actually Shapes a Response · Equation 6

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Equation guide → · This term → · Article →

r(x,y)

Claude, From First Principles: Training, Constitutional Methods, and What Actually Shapes a Response · Equation 7

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Equation guide → · This term → · Article →

r(x,y)

Claude, From First Principles: Training, Constitutional Methods, and What Actually Shapes a Response · Equation 10

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Equation guide → · This term → · Article →

C_{\mathrm{RAG}}(Q) = p_{\mathrm{store}} \cdot S \;+\; Q \cdot \left( p_{\mathrm{ru}} \cdot r(S) + p_{\mathrm{rerank}} \right)

The Hidden Infrastructure Bill Behind Every RAG Answer · Equation 1

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Equation guide → · This term → · Article →

r(S)

The Hidden Infrastructure Bill Behind Every RAG Answer · Equation 3

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Equation guide → · This term → · Article →

1/r

Measuring What a RAG System Retrieves, Not Just What It Answers · Equation 9

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Equation guide → · This term → · Article →

r

Measuring What a RAG System Retrieves, Not Just What It Answers · Equation 10

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Equation guide → · This term → · Article →

r(d)

What Actually Happens Between a Query and an Answer in RAG · Equation 5

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Equation guide → · This term → · Article →

\mathrm{RRF}(d) = \sum_{r \in R} \frac{1}{k + r(d)}

What Actually Happens Between a Query and an Answer in RAG · Equation 8

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Equation guide → · This term → · Article →

\mathrm{RRF}(d) = \sum_{r \in R} \frac{1}{k + r(d)},

Naive, Graph, and Agentic: A Systems Comparison of RAG Architectures · Equation 14

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Equation guide → · This term → · Article →

r

The Logistics Tail: Sustainment as the Binding Constraint on Organised Force · Equation 5

the radius.

Equation guide → · This term → · Article →

2r/v

The Logistics Tail: Sustainment as the Binding Constraint on Organised Force · Equation 6

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Equation guide → · This term → · Article →

P(r) = L_0 - \frac{2rf}{v}, \qquad r_{\max} = \frac{v\,L_0}{2f}.

The Logistics Tail: Sustainment as the Binding Constraint on Organised Force · Equation 7

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Equation guide → · This term → · Article →

T(r)

The Logistics Tail: Sustainment as the Binding Constraint on Organised Force · Equation 11

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Equation guide → · This term → · Article →

r

The Logistics Tail: Sustainment as the Binding Constraint on Organised Force · Equation 12

the distance.

Equation guide → · This term → · Article →

\frac{dS}{dt} = T(r) - C, \qquad T(r_c) = C .

The Logistics Tail: Sustainment as the Binding Constraint on Organised Force · Equation 13

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Equation guide → · This term → · Article →

T(r)

The Logistics Tail: Sustainment as the Binding Constraint on Organised Force · Equation 16

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Equation guide → · This term → · Article →

\forall V.\ \operatorname{packing}(V) \implies \exists c.\ \forall r.\ 1 \le r \implies \operatorname{card}\!\left(V \cap B(0,r)\right) \le \frac{\pi r^{3}}{\sqrt{18}} + c\,r^{2}

What a Proof Assistant Changes · Equation 1

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

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U(r,\theta) = \frac{GM}{r}\left[1 - \sum_{n=2}^{\infty} J_{n}\left(\frac{R}{r}\right)^{n} P_{n}(\cos\theta)\right]

How We Know What Is Inside a Planet · Equation 7

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Equation guide → · This term → · Article →

(R/r)^{n}

How We Know What Is Inside a Planet · Equation 8

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Equation guide → · This term → · Article →

c = p_{\mathrm{in}} n + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}} \left( y + z \right)

The Unit Economics of a Token · Equation 1

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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r

The Unit Economics of a Token · Equation 4

the five counts in that order and each p is the corresponding published rate.

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E_{q} = \bar{P} \cdot \frac{N_{\mathrm{tok}}}{r}

The Deployment Envelope: Small Models Where the Power Is Not · Equation 10

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Equation guide → · This term → · Article →

r

The Deployment Envelope: Small Models Where the Power Is Not · Equation 13

Equation guide → · This term → · Article →

r

The Deployment Envelope: Small Models Where the Power Is Not · Equation 15

Equation guide → · This term → · Article →

r

The Deployment Envelope: Small Models Where the Power Is Not · Equation 16

Equation guide → · This term → · Article →

\mathrm{RRF}(d) = \sum_{r \in R} \frac{1}{k + r(d)}

Retrieval Is an Evidence System, Not a Memory · Equation 11

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Equation guide → · This term → · Article →

rrr

Meanings in context

Real-valued model value

Symbol r

Symbol r

Symbol r

Symbol r

Symbol r

Symbol r

Symbol r

Symbol r

Symbol r

Symbol r

Used in 95 equations

$r$